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Gravitational Waveforms
from Coalescing Binary Black Holes
Dae-Il (Dale) Choi
NASA Goddard Space Flight Center, MD, USA
Universities Space Research Association, USA
Supported by NASA ATP02-0043-0056 & NASA Advanced Supercomputing Project
“Columbia”
Numerical Relativity 2005 Workshop
NASA Goddard Space Flight Center, Greenbelt, MD, NOV 2, 2005
Collaborators
Beyond Einstein: From the Big Bang to Black Holes
It’s teamwork
Joan Centrella, John Baker (NASA/GSFC)
Jim van Meter, Michael Koppitz (National Research Council)
Breno Imbiriba, W. Darian Boggs, Stefan Mendez-Diez
(University of Maryland)
Other collaborators
J. David Brown (North Carolina State Univ.)
David Fiske (DAC, formerly NASA/GSFC)
Kevin Olson (NASA/GSFC)
Numerical Relativity 2005 Workshop, NASA/GSFC, NOV 2,2005 2
Outline
Beyond Einstein: From the Big Bang to Black Holes
Methodology: Hahndol Code [Hahndol=한돌=translation of “Ein-stein” into
Korean]
Results: Inspiral merger from the ISCO (QC0)
Results: Head-on collision (if time allows)
Movie of the real part of Psi4
Numerical Relativity 2005 Workshop, NASA/GSFC, NOV 2,2005 3
Hahndol Code
Beyond Einstein: From the Big Bang to Black Holes
3+1 Numerical Relativity Code
– BSSN formalism following Imbiriba et al, PRD70, 124025 (2004), Alcubierre at
al PRD67, 084023 (2003) except the new gauge conditions.
– Uses finite differencing (mixed 2nd and 4th order FD, Mesh-Adapted-
Differencing–see posters for details), iterative Crank-Nicholson time integrator.
– Computational infrastructure based on PARAMESH (MacNiece, Olson)
Scalability shown up to 864 CPUs with ~ 95% efficiency.
Mesh refinement
– Currently use fixed mesh structure with mesh boundaries at (2,4,8,16,32,64)M
for QC0 runs.
– The innermost level contains the both black holes.
– For higher QC-sequence, AMR implementation being tested.
Numerical Relativity 2005 Workshop, NASA/GSFC, NOV 2,2005 4
Hahndol Code
Beyond Einstein: From the Big Bang to Black Holes
Outer boundary conditions
– Impose outgoing Sommerfeld conditions on all BSSN variables.
– But, basic strategy is to push OB far away so that OB does not
contaminate regions of interests.
– With OB=128M, no harmful effects on the dynamics of black holes nor
waveform extraction (QC0). If desired, OB can be put at 256M or beyond.
Initial data solver
– Uses multi-grid method on a non-uniform grid using Brown’s algorithm:
Brown & Lowe, JCP 209, 582-598, 2005 (gr-qc/0411112).
– Generate QC ID by solving HCE using puncture method (Brandt &
Bruegmann, 1997).
– Bowen-York prescription for the extrinsic curvature for binary black holes.
Numerical Relativity 2005 Workshop, NASA/GSFC, NOV 2,2005 5
Hahndol Code
Beyond Einstein: From the Big Bang to Black Holes
Traditional gauge conditions (AEI, etc.)
– Split conformal factor into time-indep. singular part (ΨBL) and time-dep.
regular part. Treat ΨBL analytically and evolve only the regular part.
– Use the following K-/Gamma-driver conditions for gauges. (BL factor)
– Problem is that, because of ΨBL factor, black holes cannot move.
– Requires co-rotation shift. But it involves superluminal shift.
Alternative gauge conditions
– Do not split into singular/regular part. No BL factor.
– Combined with the driver conditions, let the black holes move across
the grid.
– Does this really work?
Numerical Relativity 2005 Workshop, NASA/GSFC, NOV 2,2005 6
Hahndol Code
Beyond Einstein: From the Big
Not so fast! Two concerns. Bang to Black Holes
– (a) Puncture memory effect: BHs move but
still spiky errors at where the punctures were
at t=0.
– (b) Messy stuff near the would-have-been
puncture locations if they were moving.
The problem (a)
– Caused by the zero-speed mode in the
Gamma driver shift condition
– Can be alleviated by “shifting shift”
[Movies] comparison bet. (1) Traditional (crashed at t=35M)
(2) No BL factor (3) NoBL + Shifting Shift
Numerical Relativity 2005 Workshop, NASA/GSFC, NOV 2,2005 7
Hahndol Code
Beyond Einstein: From the Big Bang to Black Holes
The problem (b)
– In practice, we find that the stuff
doesn’t seem to “spill over”.
– [Movie: Head-on collision w/ L/M~9
using NoBL+Shifting Shift] shows a
good convergence of HC from 3
runs with different resolutions.
– Note, with the traditional gauge, HC
too large and non-convergent.
For all the cases we considered,
this new gauge conditions allow us
to obtain convergent results
(constraints, waveforms).
Numerical Relativity 2005 Workshop, NASA/GSFC, NOV 2,2005 8
Hahndol Code
Beyond Einstein: From the Big Bang to Black Holes
Wave extraction
– Compute the Newman-Penrose Weyl scalar Ψ4 (a gauge invariant measure)
where C is weyl tensor and (l,n,m,mbar) is a tetrad.
– Analyze its harmonic decomposition using a novel technique due to Misner
(Misner 2004; Fiske 2005).
– Compute waveforms r ~ 20M, 30M, 40M and 50M.
Coulomb scalar χ [Beetle, et al, PRD72, 024013 (2005); Burko,
Baumgarte & Beetle, gr-qc/0505028.]
Numerical Relativity 2005 Workshop, NASA/GSFC, NOV 2,2005 9
Evolution of Quasi Circular Initial Data
Beyond Einstein: From the Big Bang to Black Holes
QC-sequence (Minimization of effective potential, Cook 1994)
QC0, L/M=4.99, J/M^2=0.779
Re-Coulomb invariant: ReC(horizon) = -1/(8M2) for quiecent BHs. [Movie: Horizon at
ReC~ -1/2 (yellowish) at T=0; Horizon at ReC~-1/8 (blue edge) late times.]
4M 180M
Numerical Relativity 2005 Workshop, NASA/GSFC, NOV 2,2005 10
QC0 (BH source region)
Beyond Einstein: From the Big Bang to Black Holes
Comparison of Re (Coulomb) scalar for three different resolutions: M/16,
M/32, M/48 runs. [Only in this movie, time label is in terms of (M/2)]
(In this talk, different runs are labeled by the resolution in the finest
resolution grid.)
Numerical Relativity 2005 Workshop, NASA/GSFC, NOV 2,2005 11
QC0 (BH source region)
Beyond Einstein: From the Big Bang to Black Holes
Convergence of HC near black holes along x-axis from M/24 (Dashed)
and M/32 (Solid) runs. Data from Time=11M,19M,24M where BHs are
crossing the x-axis. (Note FMR boundaries are at 2M, 4M, etc.)
Numerical Relativity 2005 Workshop, NASA/GSFC, NOV 2,2005 12
QC0 Waveforms
Beyond Einstein: From the Big Bang to Black Holes
Waveforms (Re L=2, M=2 mode) from three runs, M/16, M/24, M/32
extracted at rextract =20M (Solid), 40M(Dashed). Plotted are (r x Psi4).
Good O(1/r) propagation behavior; M/24, M/32 are very close.
Comparison with Lazarus I
--Baker et al, PRD 65,124012 (2002)
Numerical Relativity 2005 Workshop, NASA/GSFC, NOV 2,2005 13
QC0 (Waveforms)
Beyond Einstein: From the Big Bang
Convergence of waveformsto Black Holes imaginary parts of L=2, M=2
(real and
mode) at r=20M (upper panels), and 40M (lower panels).
Numerical Relativity 2005 Workshop, NASA/GSFC, NOV 2,2005 14
QC0 (dE/dt, dJz/dt)
Beyond momentum loss Bang to Black
Energy & angularEinstein: From the Bigdue to GW Holes
dE/dt, dJz/dt
Numerical Relativity 2005 Workshop, NASA/GSFC, NOV 2,2005 15
QC0 (Energy and Angular momentum)
Beyond Einstein: From the Big Bang to Black Holes
Total E and Total Jz loss (plotted
for three resolutions and for 4
different extraction radii)
At r=30M,
Resolution E Jz
M/16 0.0494 -0.200 (26%)
M/24 0.0325 -0.133 (17%)
M/32 0.0315 -0.132 (17%)
Lazarus I 0.025 -0.093 (12%)
Final J~0.65
Numerical Relativity 2005 Workshop, NASA/GSFC, NOV 2,2005 16
QC0 (Energy Conservation?)
Beyond Einstein: From the Big Bang to & York,
Calculate ADM Mass (Murchadha Black Holes 1974)
Energy conservation: Minit-Mfinal= EGW?
r=40M,50M, Solid represents M(t),
Dashed M(t=0)-EGW(t).
Minit-Mfinal= EGW!
Numerical Relativity 2005 Workshop, NASA/GSFC, NOV 2,2005 17
Head-On Collision
Beyond Einstein: From the Big Bang to Black Holes
Left Panel: Waveforms extracted at rextract = 20M, 30M, 40M, 50M
– Colored lines show O(1/r) propagation fall-off behavior (M1=M2=0.5)
Right Panel: dE/dt (total energy loss ~ 0.00040)
Numerical Relativity 2005 Workshop, NASA/GSFC, NOV 2,2005 18
Head-On Collision
Beyond Einstein: From the Big Bang to Black Holes
Left Pane: Waveforms in different resolutions. Proper separation ~9M
Right Panel: convergence behavior of the waveforms.
No apparent problems up to L~11-12M. Promising for collision with large
initial separation
Numerical Relativity 2005 Workshop, NASA/GSFC, NOV 2,2005 19
